Suppose the contrary. Then \(e =n-2\)  by Lemmas REF  and REF , and hence \(X\cong P\Omega _{n}^{+}(q)\) . By Proposition REF , \(X_{B}^{^}\)  lies in a parabolic subgroup of type \(P_{2}\)  or \(P_{1,n-1}\)  of \(SL_{n}(q)\) . If the former occurs, then \(X_{B}^{^}\) 
preserves a non-degenerate 2-dimensional subspace of \(V_{n}(q)\)  of type \(-\)  by [1]}, Propositions 4.1.6 and 4.1.20, since \(n\geqslant 6\)  by Remark REF .
If \(X_{B}^{^}\)  lies in a parabolic subgroup of
type \(P_{1,n-1}\)  of \(SL_{n}(q)\) , then \(X_{B}^{^}\)  preserves
two subspaces \(U\) , \(W\)  of \(V_{n}(q)\)  such that \(U<W\)  and \(\dim U=1\)  and \(\dim W=n-1\) . Also, \(U\)  is non-degenerate by [1]}, Proposition 4.1.20,
since \(\Phi _{(n-2)f}^{\ast }(q)^{1/2}\)  divides the order of \(X_{B}^{^}\) . Thus \(W^{\perp }\ne U\) , since \(W^{\perp }<U^{\perp }\) , and hence \(X_{B}^{^}\)  preserves \(W^{\perp }\)  with \(\dim W^{\perp }=1\) . Then \(W^{\perp }\oplus U\)  is a non-degenerate 2-dimensional subspace of \(V_{n}(q)\)  of type \(-\)  preserved by \(X_{B}^{^}\) . Therefore, \(X_{B}^{^}\) 
preserves a non-degenerate 2-dimensional subspace of \(V_{n}(q)\)  of type \(-\)  in each
case.
The group \(X_{B}^{^}\)  is contained in a maximal subgroup \(M\)  of \(X^{^}\) , where \(M\)  is either \(\left( Z_{\frac{q+1}{(q-1,2)}}\times \Omega _{n-2}^{-}(q)\right) .Z_{2}\)  or \(\left( Z_{\frac{q+1}{(q-1,2)}}\times \Omega _{n-2}^{-}(q)\right) .[4]\)  according to whether \(q\)  is
even or odd respectively, by [1]}, Propositions 4.1.6. Thus
\(\left|X_{B}\right|_{p}=\left|X_{B}^{^}\right|_{p}=(2,q)\left|X_{B}^{^}\cap \Omega _{n-2}^{-}(q)\right|_{p}.\) 
Note that, \(\Phi _{(n-2)f}^{\ast }(q)^{1/2}\)  divides the order of \(X_{B}^{^}\cap \Omega _{n-2}^{-}(q)\)  and the index of it in \(SL_{n-2}(q)\) .
Also \(p\mid \left|X_{B}^{^}\right|\)  by Corollary REF , and hence \(p\)  divides the order of \(X_{B}\cap \Omega _{n-2}^{-}(q)\)  by (REF ), since \(X \ncong P\Omega _{6}(2)^{+}\)  by Lemma REF .
Then we may apply Proposition REF , with \(X_{B}\cap \Omega _{n-2}^{-}(q)\) 
in the role of \(Y\)  and \(SL_{n-2}(q)\)  in the role of \(SL_{n}(q)\) , thus obtaining either \(q=7\) , \(n-2=4\)  and \(X_{B}^{^}\cap \Omega _{n-2}^{-}(q) \cong Z_{2}.A_{7}\) , or \(X_{B}^{^}\cap \Omega _{n-2}^{-}(q)\leqslant GL_{1}(q^{n-2})\cdot Z_{n-2}\)  with \(p\mid n-2\) . Since \(X\cong P\Omega _{6}^{+}(7) \cong PSL_{4}(7)\) , we may apply the same argument used in the proof of Proposition REF  to rule out this case. Thus \(X_{B}^{^}\cap \Omega _{n-2}^{-}(q)\leqslant GL_{1}(q^{n-2})\cdot Z_{n-2}\)  and
hence \(\left|X_{B}^{^}\cap \Omega _{n-2}^{-}(q)\right|_{p}=(n-2)_{p}\leqslant (n-2)\) . Therefore
\( q^{\frac{n(n-2)-12}{12}}\leqslant \left|X_{B}\right|_{p}\leqslant (2,q)(n-2)\) 
by Corollary REF (v), where \(n\geqslant 6\)  by Remark REF . It is
straightforward to see that, only \(n=6\)  and \(q=2,4\)  or 8 fulfill (REF ). However, all these cases are ruled out. Indeed, \(q=2\)  cannot occur by Lemma REF , whereas \(q=4\)  or 8
cannot occur since \(\Phi _{8}^{\ast }(2)=17\)  and \(\Phi _{12}^{\ast }(2)=13\)  respectively.
